Linear Algebra Answers

Whichofthefollowingstatementsaretrueandwhicharefalse?Justifyyouranswerwitha shortprooforacounterexample
1)Thefunctionf:R→Rdefinedbyf(x)=cosxis1-1.

LetVbethesetofallfunctionsthataretwicedifferentiableinRand S={cosx,sinx,xcosx,xsinx}. a)CheckthatSisalinearlyindependentsetoverR.(Hint:Considertheequation a0cosx+a1sinx+a2xcosx+a3xsinx. Putx=0,π,π 2 ,π 4 ,etc.andsolveforai.)

Solve the linear equation
2x+y-3z=5
3x-2y-2z=5
5x-3y-z=16

Solve the linear equation
2x+y-3z=5
3x-2y-2z=3
5x-3y-2=16

Find X by the use of determinant:
3x-4y+2z+8=0
x+5y-3z+2=0
5x+3y-z+6=0

Solve the linear equations for x,y,z
2x+y-3z=5
3x-2y-2z=3
5x-3y-2=16

Find z by the use of determinant:
3x-4y+2z+8=0
x+5y-3z+2=0
5x+3y-z+6=0

Solve the set of linear equations by Gaussian elimination method:
a+3b+2c=3
2a-b-3c=-8
5a+2b+c=9
Solve for c

Solve the set of linear equations by matrix method and Gaussian elimination method :
a+2b+3c=5
3a-b+2c=8
4a-6b-4c=-2
Solve for b

Gauss Elimination Method

0 = 0.13D1 + 0.03D2
0 = 0.3D1 + 0.13D2 - 0.1D4
2 = 0.13D3 - 0.03D4
-4= -0.1D2 - 0.03D3 + 0.13D4